Question

In Young's double slit experiment, for wavelength \( \lambda_1 \), the \( n^{\text{th}} \) bright fringe is obtained at a point \( P \) on the screen. Keeping the same setting, source of light is replaced by wavelength \( \lambda_2 \), and now \( (n+1)^th \) bright fringe is obtained at the same point \( P \) on the screen. The value of \( n \) is:

• A. \( \frac{\lambda_1 - \lambda_2}{\lambda_2} \)
• B. \( \frac{\lambda_1}{\lambda_1 - \lambda_2} \)
• C. \( \frac{\lambda_1 - \lambda_2}{\lambda_1} \)
• D. \( \frac{\lambda_2}{\lambda_1 - \lambda_2} \)

Hint:

In Young's double slit experiment, the fringe separation is inversely proportional to the slit separation and directly proportional to the wavelength. The position of the fringes depends on the wavelength of the light used.

Answer 1

The Correct Option is D

Step 1: Fringe Separation Formula.
The fringe separation \( \beta \) in a double slit experiment is given by: \[ \beta = \frac{\lambda D}{d} \] where \( \lambda \) is the wavelength of the light, \( D \) is the distance between the slits and the screen, and \( d \) is the separation between the slits. The position of the \( n^{\text{th}} \) bright fringe is given by: \[ y_n = n \beta \] For the first light with wavelength \( \lambda_1 \), the \( n^{\text{th}} \) bright fringe is at position \( y_n \). For the second light with wavelength \( \lambda_2 \), the fringe separation changes, and the \( (n+1)^{\text{th}} \) bright fringe is obtained at the same position \( P \) on the screen. By equating the two equations for \( y_n \), we find: \[ n \frac{\lambda_1 D}{d} = (n+1) \frac{\lambda_2 D}{d} \] Simplifying, we get the value of \( n \) as: \[ n = \frac{\lambda_2}{\lambda_1 - \lambda_2} \] Step 2: Final Answer.
Thus, the value of \( n \) is \( \frac{\lambda_2}{\lambda_1 - \lambda_2} \).